To determine how the graph of [tex]\( f(x) = 2^x \)[/tex] changes to become [tex]\( g(x) = 3 \cdot 2^x + 5 \)[/tex], let's analyze the transformations step by step.
1. Analyze the transformation
- The original function is [tex]\( f(x) = 2^x \)[/tex].
- The transformed function is [tex]\( g(x) = 3 \cdot 2^x + 5 \)[/tex].
2. Break down [tex]\( g(x) \)[/tex]
- [tex]\( g(x) = 3 \cdot 2^x + 5 \)[/tex] consists of two main components: [tex]\( 3 \cdot 2^x \)[/tex] and [tex]\( +5 \)[/tex].
3. Component [tex]\( 3 \cdot 2^x \)[/tex]
- This term indicates that the function [tex]\( 2^x \)[/tex] is multiplied by 3.
- Multiplying the function by 3 results in a vertical stretch. This means that every point on the graph of [tex]\( 2^x \)[/tex] will be stretched vertically away from the x-axis by a factor of 3.
4. Component [tex]\( +5 \)[/tex]
- This term indicates that the function [tex]\( 2^x \)[/tex] (after being multiplied by 3) is then shifted upward by 5 units.
- Adding 5 to the function means every point on the graph is lifted up by 5 units.
Based on this analysis, the correct transformations are:
B. It is vertically stretched.
- The multiplication by 3 causes a vertical stretch.
C. It is shifted up.
- The addition of 5 shifts the graph upwards by 5 units.
The transformations to the graph of [tex]\( f(x) = 2^x \)[/tex] to become [tex]\( g(x) = 3 \cdot 2^x + 5 \)[/tex] are vertical stretch and upward shift.
Therefore, the selected options are:
- B. It is vertically stretched.
- C. It is shifted up.
